Construction of solutions of nonlinear two-dimensional problems on current distribution in an anisotropically conducting medium: PMM vol. 38, n≗5. 1974. pp. 819–828

Abstract

Stationary two-dimensional electric current distributions in an anisotropically conducting medium having a nonlinear Ohm's law, are described by the system of equations formulated in [1], Depending on the character of the nonlinear relation between the current density and the electric field,and on the value of the Hall parameter β, this system can be of an elliptic or hyperbolic type. For β = 0 the electrodynamic equations are analogous to the equations for potential gas dynamic flows, therefore by analogy these problems can be solved by the hodograph transformation, as it is done in gas dynamics [2], The hodograph transformation generalized for the case β ≠ 0 is applied below to simple two-dimensional problems. The relation between the type of system and the positive definiteness of the symmetric part of the differential conductivity tensor, is established. Linear equations in the hodograph plane of an effective electric field are obtained for the potential and for a function of the electric current. Boundary conditions are formulated in terms of each of these functions on the image lines for the electrode and dielectric regions with straight-line boundaries. For the elliptic case the solution of two asymptotic problems are obtained and examined: 1. (1) the field distribution in a strip between a perfectly conducting wall and a dielectric wall; 2. (2) the current concentration in the region of a semi-infinite electrode edge. The possibility of corresponding solutions for the hyperbolic case is discussed. For β ≠ 0 exact solutions for particular dependence of the electrical conductivity on current density, corresponding to the hyperbolic region, are obtained by the method of characteristics used in [3, 4], There are reasons for assuming that the distribution is unstable for hyperbolic modes [1, 3– 5], For homogeneous states of a nonequilibrium plasma, the β value corresponding to a change of the system type, also determines the ionization instability limit. In the general case of nonhomogeneous states the instability of hyperbolic and mixed solutions is not proved. In the elliptic region the equations describe stable current distributions, therefore the construction of elliptic solutions is of the greatest physical interest.